Radial basis function partition of unity methods for PDEs

Transcription

1 Radial basis function partition of unity methods for PDEs Elisabeth Larsson, Scientific Computing, Uppsala University Credit goes to a number of collaborators Alfa Ali Alison Lina Victor Igor Heryudono Safdari Ramage von Sydow Shcherbakov Tominec Localized Kernel-Based Meshless Methods for Partial Differential Equations, ICERM, Aug 8, 2017 E. Larsson, Aug 8, 2017 (1 : 28)

2 RBF partition of unity methods for PDEs E. Larsson, Aug 8, 2017 (2 : 28)

3 Motivation for Global RBF approximation + Ease of implementation in any dimension. + Flexibility with respect to geometry. + Potentially spectral convergence rates. Computationally expensive for large problems. Local RBF approximations on patches are blended into a global solution using a partition of unity. Provides spectral or high-order convergence. Solves the computational cost issues. Allows for local adaptivity. Wendland (2002), Fasshauer (2007), Cavoretto, De Rossi, Perracchione et al., Larsson, Heryudono et al. E. Larsson, Aug 8, 2017 (3 : 28)

4 The RBF partition of unity method Global approximation P ũ(x) = w j (x)ũ j (x) j=1 PU weight functions Generate weight functions from compactly supported C 2 Wendland functions ψ(ρ) = (4ρ + 1)(1 ρ) 4 + using Shepard s method w i (x) = ψ i (x) M j=1 ψ j (x). Ω j Cover Each x Ω must be in the interior of at least one Ω j. Patches that do not contain unique points are pruned. E. Larsson, Aug 8, 2017 (4 : 28)

5 Differentiating approximations Applying an operator globally M ũ = w i ũ i + 2 w i ũ i + w i ũ i i=1 Local differentiation matrices Let u i be the vector of nodal values in patch Ω i, then u i = Aλ i, where A ij = φ j (x i ) Lu i = A L A 1 u i, where A L ij = Lφ j (x i ). The global differentiation matrix Local contributions are added into the global matrix. 300 E. Larsson, Aug 8, 2017 (5 : 28) nz = 10801

6 An collocation method Choices & Implications Nodes and evaluation points coincide. Square matrix, iterative solver available (Heryudono, Larsson, Ramage, von Sydow 2015). Global node set. Solutions ũ i (x k ) = ũ j (x k ) for x k in overlap regions. Patches are cut by the domain boundary. Potentially strange shapes and lowered local order. E. Larsson, Aug 8, 2017 (6 : 28)

7 An least squares method Choices & Implications Each patch has an identical node layout. Computational cost for setup is drastically reduced. Evaluation nodes are uniform. Easy to generate both local and global high quality node sets. Patches have nodes outside the domain. Good for local order, but requires denser evaluation points. E. Larsson, Aug 8, 2017 (7 : 28)

8 The interpolation error E α = D α (I (u) u) = M j=1 β α The weight functions For C k weight functions and α k D α w j L (Ω j ) C α H α j ( ) α D β w j D α β (I (u j ) u j ) β, H j = diam(ω j ). The local RBF interpolants (Gaussians) Define the local fill distance h j (Rieger, Zwicknagl 2010) D α (I (u j ) u j ) L ( Ω j ) c α,jh m j d 2 α j u j N ( Ω j ), D α (I (u j ) u j ) L ( Ω j ) eγ α,j log(h j )/ h j u j N ( Ωj ). E. Larsson, Aug 8, 2017 (8 : 28)

9 interpolation error estimates Algebraic estimate for H j /h j = c E α L (Ω) K max 1 j M C j H m j d 2 α j u N ( Ωj ) K Maximum # of Ω j overlapping at one point m j Related to the local # of points Ω j Ω j Ω Spectral estimate for fixed partitions E α L (Ω) K max 1 j M Ce γ j log(h j )/ h j u N ( Ωj ) Implications Bad patch reduces global order. Two refinement modes. Guidelines for adaptive refinement. E. Larsson, Aug 8, 2017 (9 : 28)

10 Error estimate for PDE approximation The PDE estimate ũ u L (Ω) C P E L + C P L,X L + Y,X (C M δ M + E L ), where C P is a well-posedness constant and C M δ M is a small multiple of the machine precision. Implications Interpolation error E L provides convergence rate. Norm of inverse/pseudoinverse can be large. Matrix norm better with oversampling. Finite precision accuracy limit involves matrix norm. Follows strategies from Schaback (2007) and Schaback (2016) E. Larsson, Aug 8, 2017 (10 : 28)

11 Does require stable methods? In order to achieve convergence we have two options Refine patches such that diameter H decreases. Increase node numbers such that N j increases. In both cases, theory assumes ε fixed. The effect of patch refinement H = 1, ε = 4 H = 0.5, ε = 4 H = 0.25, ε = H H H The RBF QR method: Stable as ε 0 for N 1 Effectively a change to a stable basis. Fornberg, Piret (2007), Fornberg, Larsson, Flyer (2011), Larsson, Lehto, Heryudono, Fornberg (2013) E. Larsson, Aug 8, 2017 (11 : 28)

12 Effects on the local matrices Local contribution to a global Laplacian L j = (Wj A j + 2Wj A j + W j A j )A 1 j. Typically: A j ill-conditioned, L j better conditioned. for accuracy Stable for small RBF shape parameters ε Change of basis à = AQR1 T D1 T Same result in theory à L à 1 = A L A 1 More accurate in practice Relative error in A j A 1 j without N=10 N=20 N= ε E. Larsson, Aug 8, 2017 (12 : 28)

13 Poisson test problems in 2-D Domain Ω = [ 2, 2] 2. Uniform nodes in the collocation case. u R (x, y) = 1 25x 2 +25y 2 +1 u T (x, y) = sin(2(x 0.1) 2 ) cos((x 0.3) 2 )+sin 2 ((y 0.5) 2 ) E. Larsson, Aug 8, 2017 (13 : 28)

14 Error results with and without RBF QR Error Least squares Fixed shape ε = 0.5 or scaled such that εh = c Left: 5 5 patches Spectral mode RBF QR Direct Scaled Points per dimension Right: 55 points per patch Error Algebraic mode p= 7 RBF QR Direct Scaled p= Patches per dimension With RBF QR better results for H/h large. Scaled approach good until saturation. E. Larsson, Aug 8, 2017 (14 : 28)

15 as a function of patch size Runge Trig Error H Error H Collocation (dashed lines) and Least Squares (solid lines). Points per patch n = 28, 55, 91. Theoretical rates p = 4, 7, 10. Numerical rates p 3.9, 6.9, 9.8. E. Larsson, Aug 8, 2017 (15 : 28)

16 Spectral convergence for fixed patches Error Runge /h Error Trig /h Collocation (dashed lines) and Least Squares (solid lines). LS-RBF-PU is significantly more accurate due to the constant number of nodes per patch. E. Larsson, Aug 8, 2017 (16 : 28)

17 and large scale problems The global error estimate ũ u L (Ω) C P E L + C P L,X L + Y,X (C M δ M + E L ) The dark horse is the stability matrix norm The stability norm is related to conditioning. In the collocation case, L 1 X,X grows with N. How does it behave with least squares? E. Larsson, Aug 8, 2017 (17 : 28)

18 The stability matrix norm: fill distance Fixed patch size with patches. Oversampling M/N = 1.1, 1.2, 1.5 Stability norm /h Error /h Collocation (dashed) and Least Squares (solid) Stability norm grows exponentially as h decreases. Oversampling reduces the norm. Collocation is less robust. E. Larsson, Aug 8, 2017 (18 : 28)

19 Stability norm: Patch size Fixed number of points per patch n = 28, 55, 91 Results as a function of patch diameter H Stability(H) Error(H) Stability norm H Error H Collocation (dashed) and LS (solid) The norm does not grow for LS- (!) E. Larsson, Aug 8, 2017 (19 : 28)

20 Oversampling provides stability (at a cost). For robustness in h, M/N needs to grow with N. E. Larsson, Aug 8, 2017 (20 : 28) Stability norm: Oversampling Fixed number of local points n = 55. Patches: 5 5, 10 10, Stability norm Stability(M/N) β Collocation (dashed) and LS (solid)

21 Poisson test problems in 3-D ( ) u(x) = sin π(x1 0.5)x 3 log(x 2 +3) 1 0 E. Larsson, Aug 8, 2017 (21 : 28)

22 in 3-D Error n = 20, 35, 56, 84, 120, 165 points per patch Sphere Pepper Spectral p = 1.7 p = 4.1 p = 5.4 p = 6.6 p = 6.1 p = H Error p = 1.5 p = 3.6 p = 3.9 p = 6.2 p = 5.6 p = H Error /h order is a bit better than expected. Refinement modes work as expected. E. Larsson, Aug 8, 2017 (22 : 28)

23 Computational time comparison in 2-D Time Collocation LS Error Experiments with fixed n = 28, 55, 91. Time Error For a fixed problem size, RBF-PU-LS involves more work, but yields a smaller error. Overall, the LS approach is the winner. Strategy: Points per patch adjusted to tolerance. E. Larsson, Aug 8, 2017 (23 : 28)

24 Solution using iterative methods So far implemented for the collocation approach. First step: Re-ordering of the unknowns to improve structure. Patches: Preceded and followed by a neighbour. Nodes xk : Define home patch Ωj such that wj wi (xk ). Within patch: Sub-order according to node memberships. Heryudono, Larsson, Ramage, and von Sydow (2015) E. Larsson, Aug 8, 2017 (24 : 28)

25 Matrix structures with snake order I The patch order fills in gaps in the band. I The node order minimizes the central band width. I Chosen preconditioner, ILU(0) of central band. Structure 2-D Structure 3-D nz = E. Larsson, Aug 8, 2017 (25 : 28) 300

26 Results for the iterative method contd. Results in 2-D with Halton nodes N # it no prec # it ILU(0) Time gain Results in 3-D with Halton nodes N # it no prec # it ILU(0) Time gain Memory gain not to be forgotten. E. Larsson, Aug 8, 2017 (26 : 28)

27 Current project: With Alfa Heryudono, Danilo Stocchino and Stefano De Marchi Box structure helps book keeping. Overlaps complicate things. Evaluation points placed using node placing algorithm. Fornberg & Flyer (2015) E. Larsson, Aug 8, 2017 (27 : 28)

28 Results is a flexible tool for solving PDEs. By using template patches, LS becomes computationally efficient. LS is numerically stable for large scale problems. Things to do Change patch type to reduce unnecessary overlap. Adaptive algorithms based on LS. Time-stability for LS. For collocation and hyperbolic PDEs on the sphere, see poster by Igor Tominec et al. E. Larsson, Aug 8, 2017 (28 : 28)